Question

A uniformly charged ring of radius $3a$ and total charge $q$ is placed in xy-plane centred at origin. A point charge $q$ is moving towards the ring along the z-axis and has speed $u$ at $z=4a$. The minimum value of $u$ such that it crosses the origin is :

• A. $\sqrt{\frac{2}{m}}{\left(\frac{1}{15}\frac{q^2}{4\pi {\epsilon }_{0}a}\right)}^{1/2}$
• B. $\sqrt{\frac{2}{m}}{\left(\frac{2}{15}\frac{q^2}{4\pi {\epsilon }_{0}a}\right)}^{1/2}$
• C. $\sqrt{\frac{2}{m}}{\left(\frac{2}{15}\frac{q^2}{3\pi {\epsilon }_{0}a}\right)}^{1/2}$
• D. $\sqrt{\frac{2}{m}}{\left(\frac{1}{5}\frac{q^2}{4\pi {\epsilon }_{0}a}\right)}^{1/2}$

Answer 1

Answer: (B)

$U_i+K_i=U_f+K_f$
$\frac{k{q}^2}{\sqrt{16a^2+9a^2}}+\frac{1}{2}m{v}^2=\frac{k{q}^2}{3a}$
$\frac{1}{2}m{v}^2=\frac{k{q}^2}{a}\left(\frac{1}{3}-\frac{1}{5}\right)=\frac{2k{q}^2}{15a}$
$v=\sqrt{\frac{4k{q}^2}{15ma}}$